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Theorem funtransport 32436
Description: The TransportTo relationship is a function. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funtransport Fun TransportTo

Proof of Theorem funtransport
Dummy variables 𝑚 𝑛 𝑝 𝑞 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reeanv 3237 . . . . . 6 (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) ↔ (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
2 simp1 1130 . . . . . . . . . . 11 ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) → 𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)))
3 simp1 1130 . . . . . . . . . . 11 ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) → 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)))
42, 3anim12i 591 . . . . . . . . . 10 (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞))) → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
54anim1i 593 . . . . . . . . 9 ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞))) ∧ (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
65an4s 904 . . . . . . . 8 ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
7 xp1st 7357 . . . . . . . . . 10 (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) → (1st𝑝) ∈ (𝔼‘𝑛))
8 xp1st 7357 . . . . . . . . . 10 (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) → (1st𝑝) ∈ (𝔼‘𝑚))
9 axdimuniq 25984 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ (1st𝑝) ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ (1st𝑝) ∈ (𝔼‘𝑚))) → 𝑛 = 𝑚)
10 fveq2 6344 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (𝔼‘𝑛) = (𝔼‘𝑚))
1110riotaeqdv 6767 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))
1211eqeq2d 2762 . . . . . . . . . . . . . . 15 (𝑛 = 𝑚 → (𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ↔ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))))
1312anbi2d 742 . . . . . . . . . . . . . 14 (𝑛 = 𝑚 → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
14 eqtr3 2773 . . . . . . . . . . . . . 14 ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)
1513, 14syl6bir 244 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
169, 15syl 17 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ (1st𝑝) ∈ (𝔼‘𝑛)) ∧ (𝑚 ∈ ℕ ∧ (1st𝑝) ∈ (𝔼‘𝑚))) → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
1716an4s 904 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) ∧ ((1st𝑝) ∈ (𝔼‘𝑛) ∧ (1st𝑝) ∈ (𝔼‘𝑚))) → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦))
1817ex 449 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((1st𝑝) ∈ (𝔼‘𝑛) ∧ (1st𝑝) ∈ (𝔼‘𝑚)) → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)))
197, 8, 18syl2ani 691 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) → ((𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) → 𝑥 = 𝑦)))
2019impd 446 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))) ∧ (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
216, 20syl5 34 . . . . . . 7 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → ((((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
2221rexlimivv 3166 . . . . . 6 (∃𝑛 ∈ ℕ ∃𝑚 ∈ ℕ (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
231, 22sylbir 225 . . . . 5 ((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
2423gen2 1864 . . . 4 𝑥𝑦((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦)
25 eqeq1 2756 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)) ↔ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))))
2625anbi2d 742 . . . . . . 7 (𝑥 = 𝑦 → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
2726rexbidv 3182 . . . . . 6 (𝑥 = 𝑦 → (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
2810sqxpeqd 5290 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑚) × (𝔼‘𝑚)))
2928eleq2d 2817 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
3028eleq2d 2817 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚))))
3129, 303anbi12d 1541 . . . . . . . 8 (𝑛 = 𝑚 → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ↔ (𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞))))
3231, 12anbi12d 749 . . . . . . 7 (𝑛 = 𝑚 → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
3332cbvrexv 3303 . . . . . 6 (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))))
3427, 33syl6bb 276 . . . . 5 (𝑥 = 𝑦 → (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))))
3534mo4 2647 . . . 4 (∃*𝑥𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ↔ ∀𝑥𝑦((∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝))) ∧ ∃𝑚 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ 𝑞 ∈ ((𝔼‘𝑚) × (𝔼‘𝑚)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑦 = (𝑟 ∈ (𝔼‘𝑚)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))) → 𝑥 = 𝑦))
3624, 35mpbir 221 . . 3 ∃*𝑥𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))
3736funoprab 6917 . 2 Fun {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))}
38 df-transport 32435 . . 3 TransportTo = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))}
3938funeqi 6062 . 2 (Fun TransportTo ↔ Fun {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st𝑞) ≠ (2nd𝑞)) ∧ 𝑥 = (𝑟 ∈ (𝔼‘𝑛)((2nd𝑞) Btwn ⟨(1st𝑞), 𝑟⟩ ∧ ⟨(2nd𝑞), 𝑟⟩Cgr𝑝)))})
4037, 39mpbir 221 1 Fun TransportTo
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wal 1622   = wceq 1624  wcel 2131  ∃*wmo 2600  wne 2924  wrex 3043  cop 4319   class class class wbr 4796   × cxp 5256  Fun wfun 6035  cfv 6041  crio 6765  {coprab 6806  1st c1st 7323  2nd c2nd 7324  cn 11204  𝔼cee 25959   Btwn cbtwn 25960  Cgrccgr 25961  TransportToctransport 32434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-z 11562  df-uz 11872  df-fz 12512  df-ee 25962  df-transport 32435
This theorem is referenced by:  fvtransport  32437
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